Specific Heat Critical Exponent Research Articles

Measurement of random-exchange critical behavior in the mixed Ising system:FexCo1−xF2

We report on an observation of random-exchange (RE) critical behavior in a mixed three-dimensional Ising system. Using capacitance measurements, we have determined the critical-behavior universality class and the N\'eel temperature, ${\mathit{T}}_{\mathit{N}}$(x), as a function of concentration x of the mixed magnetic system ${\mathrm{Fe}}_{\mathit{x}}$${\mathrm{Co}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{F}}_{2}$. The specific-heat critical exponent \ensuremath{\alpha}=-0.09\ifmmode\pm\else\textpm\fi{}0.06 and the amplitude ratio ${\mathit{A}}_{+}$/${\mathit{A}}_{\mathrm{\ensuremath{-}}}$=1.6\ifmmode\pm\else\textpm\fi{}0.4, are in agreement with previous experimental results on magnetically diluted (e.g., ${\mathrm{Fe}}_{\mathit{x}}$${\mathrm{Zn}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{F}}_{2}$) RE systems. A mean-field interpretation of ${\mathit{T}}_{\mathit{N}}$(x) vs x yields an Fe-Co exchange interaction ${\mathit{J}}_{\mathrm{Co}\mathrm{\ensuremath{-}}\mathrm{Fe}}$=13.10\ifmmode\pm\else\textpm\fi{}0.30 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$.

Evaluation of the correlation length critical exponent of the N‐vector model

AbstractAn extension of a heuristic treatment of the critical behavior of the Ising model (N = 1) developed by Thompson, to the general N‐vector model is made. Two approximations for the correlation length critical exponent v as a function of the lattice spatial dimension d and of the number of components of the order parameter N are obtained. Using the hyperscaling relation, the specific heat critical exponent α is also evaluated. Finally, diagrams are constructed displaying in the plane (d, N) various regions, namely: classical critical indexes, non‐classical critical indexes, and no‐phase transition regions or regions where the critical temperature goes to zero. These diagrams are also used to separate regions with divergent specific heat (α > 0) from the ones where it has a cusp‐like behavior (α < 0).

Asymptotic and leading correction-to-scaling specific-heat critical exponents and amplitudes for quench-disordered ferromagnets from resistivity measurements.

We report an experimental determination of the asymptotic and leading correction-to-scaling specific-heat critical exponents and amplitudes for quench-disordered ferromagnets. The renormalization-group estimates, available for some of the amplitude ratios only, are in agreement with the presently determined values. The scaling relation ${\mathrm{\ensuremath{\alpha}}}^{+}$=${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}}$ is obeyed and the asymptotic amplitude ratio ${\mathit{A}}^{+}$/${\mathit{A}}^{\mathrm{\ensuremath{-}}}$ possesses a value characteristic of a three-dimensional isotropic nearest-neighbor Heisenberg ferromagnet with isotropic long-range dipolar interactions.

Critical properties of a dilute gas of vortex rings in three dimensions and the lambda transition in liquid helium.

We compute the critical properties of a dilute gas of circular and elliptical vortex rings in a three-dimensional superfluid using a scale-dependent mean-field approximation of the Kosterlitz-Thouless type. A set of scaling differential equations is derived and solved numerically. The system has a second-order phase transition with superfluid density critical exponent \ensuremath{\nu}=0.57 and specific-heat critical exponent \ensuremath{\alpha}=0.64 for circular rings and 0.50<\ensuremath{\nu}<0.53, 0.94<\ensuremath{\alpha}<0.99 for elliptical rings of small eccentricity. The relevance of these results for the understanding of the \ensuremath{\lambda} transition in liquid helium is discussed.

Specific-heat exponent and critical-amplitude ratio at the Lifshitz multicritical point.

The specific-heat critical exponent, ${\ensuremath{\alpha}}_{L}$, associated with the uniaxial Ising-type Lifshitz point (LP) in MnP, was determined experimentally from the study of the magnetic susceptibility, \ensuremath{\chi}, perpendicular to the easy axis, as a function of the magnetic field. According to generalized scaling theory, the divergence of \ensuremath{\chi}, at the ferromagnetic-paramagnetic phase boundary, near the LP, is governed by \ensuremath{\alpha}. We found ${\ensuremath{\alpha}}_{L}$ between 0.4 and 0.5, in agreement with estimates from other experimentally known critical exponents in MnP. For the critical-amplitude ratio we found ${A}^{+}$/${A}^{\mathrm{\ensuremath{-}}}$=0.65\ifmmode\pm\else\textpm\fi{}0.05. .AE

Spin dependence of effective critical exponent of specific heat for the high-spin two-dimensional system

An expression of the effective specific-heat critical exponent with an apparent spin dependence for the general isotropic two-dimensional system is given by the conformal theory. This expression may be described as a critical behavior for the high-spin two-dimensional system away from the critical point.

Potts ferromagnet: Transformations and critical exponents in planar hierarchical lattices

We prove that the duality transformation for a Potts ferromagnet on two-rooted planar hierarchical lattices (HL) preserves the thermal eigenvalue. This leads to a relation between the correlation length critical exponents υ of a HL and its corresponding dual lattice. Using hyperscaling, we show that their specific heat critical exponents α coincide. For a smaller class of HL—namely of diamond and tress types—we prove that another transformation also preserves υ and α.

Spin-glass in low dimensions and the Migdal-Kadanoff approximation

Abstract We study the spin-glass problem within the Migdal-Kadanoff approximation of the hyper cubic lattices. Using various approaches, both analytical and numerical, we perform the real space renormalization of the problem. We find that a spin-glass transition occurs in three dimensions while it does not occur in two dimensions. The specific heat critical exponent for the transition is found to be large and negative, in agreement with the experimental results.

Specific-heat anomaly of Au(110) (1 × 2) studied by low-energy electron diffraction

Abstract : The specific heat critical exponent alpha has been measured for the Gold (110) (1X2) order disorder phase transition using partially integrated Low Energy Electron Diffraction (LEED) intensities. The resulting value, alpha = 0.02 + or - 0.05, is consistent with the predicted using universality class of this transition. Evidence for a reduction in the effective critical temperature due to finite size effects is also presented. Keywords: Phase transitions; Universality; Critical exponents; Low energy electron-diffraction; and Finite size effects.

Specific-heat study of random-field and competing-anisotropy effects in Fe1-xCoxCl2.

We report a comprehensive specific-heat study of the mixed antiferromagnetic system ${\mathrm{Fe}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Co}}_{\mathrm{x}}$${\mathrm{Cl}}_{2}$ with x=0.286, 0.366, and 0.604. Four different issues related to competing anisotropies and random fields are discussed. (i) In zero field, a single sharp peak at the N\'eel transition ${T}_{N}$ is seen in each sample. There is no evidence for additional transitions at lower temperatures. We suggest that there are differences between systems with competing Ising-XY anisotropies and those with Ising-Ising anisotropies. Some new neutron scattering data are presented.The behavior of other competing anisotropy systems is discussed. (ii) When a small magnetic field H is applied along the easy axis (c axis) in ${\mathrm{Fe}}_{0.714}$${\mathrm{Co}}_{0.286}$${\mathrm{Cl}}_{2}$, it exhibits a random-field Ising model (RFIM) behavior similar to the diluted system ${\mathrm{Fe}}_{0.682}$${\mathrm{Mg}}_{0.318}$${\mathrm{Cl}}_{2}$.The reduction of ${T}_{N}$ obeys the scaling prediction ${T}_{N}$(H)=${T}_{N}$(0)-${\mathrm{AH}}^{2}$-${\mathrm{BH}}^{2/\ensuremath{\varphi}}$, where the crossover exponent \ensuremath{\varphi} is found to be 1.24\ifmmode\pm\else\textpm\fi{}0.09. The shape of the specific-heat peak also changes continuously with increasing field. This is attributed to crossover effects in constant applied fields. Small thermal hystereses between the field-cooled and field-warmed data were detected in ${\mathrm{Fe}}_{0.682}$${\mathrm{Mg}}_{0.318}$${\mathrm{Cl}}_{2}$, but not in ${\mathrm{Fe}}_{0.714}$${\mathrm{Co}}_{0.286}$${\mathrm{Cl}}_{2}$.Some previously unpublished results on ${\mathrm{Fe}}_{0.682}$${\mathrm{Mg}}_{0.318}$${\mathrm{Cl}}_{2}$ are presented. The measurement of the specific-heat critical exponent \ensuremath{\alpha} is also discussed. We point out that the standard indirect methods, such as susceptibility and birefringence, may be invalid in finite fields. Our direct specific-heat data suggest that \ensuremath{\alpha} is large and negative (\ensuremath{\approxeq}-1). This can be interpreted as a Fisher renormalization effect. (iii) In higher fields, ${\mathrm{Fe}}_{0.714}$${\mathrm{Cl}}_{0.286}$${\mathrm{Cl}}_{2}$ has a spin-flop (SF) phase.Sharp cusps are observed at the transitions to the paramagnetic (PM) phase, but no anomalies are observed at the transitions to the low-field uniaxial antiferromagnetic (AF) phase. The AF-PM boundary is found to join smoothly to the SF-PM boundary at an inflection point and the peak near this point shows substantial rounding. Possible experimental and theoretical causes for these observations are discussed. (iv) In ${\mathrm{Fe}}_{0.396}$${\mathrm{Co}}_{0.604}$${\mathrm{Cl}}_{2}$, the spins order perpendicular to the c axis in zero field. For applied fields parallel to the c axis, the system should correspond to the three-state Potts model in random fields. We find that the shape of the specific-heat peak changes with increasing field in a manner similar to the RFIM systems, becoming quite symmetric at 19.2 kOe. The phase boundary can be described either by the singular equation given in (ii) with an unusually small crossover exponent \ensuremath{\varphi}=0.42\ifmmode\pm\else\textpm\fi{}0.03, or by an analytic equation involving unusually large ${H}^{4}$ and ${H}^{6}$ terms. These results are not well understood.

Phases and phase transitions of para-hydrogen monolayers adsorbed on graphite

Detailed specific heat measurements of para-hydrogen physisorbed on graphite (Grafoil) have been performed at about 70 different coverages in the total density range below monolayer completion and at temperatures between 3 and 40 K. From the data a complete phase diagram is proposed showing several novel features, e.g., a new intermediate solid and a new reentrant fluid phase, a tricritical and a multicritical point. Analysing the specific heat critical exponent of the order-disorder transition of the commensurate √3 phase, a value of α = 0.33 ± 0.03 was found confirming the three-state Potts model. Interestingly, the phase diagram turns out to have striking analogies to that of other two-dimensional quantum system: 3He and 4He.

Universality of the critical behaviour of the weakly disordered Baxter model

The two-point spin-spin correlation function of the weakly disordered Baxter model in the phase transition point is calculated. The corresponding critical exponent appears to be zero, i.e. the decay of the correlation function is slower than the power law: R- eta . Together with the previous result that the specific heat critical exponent of the weakly disordered Baxter model is also universal: alpha -0, it gives the complete set of universal critical exponents which describe the phase transition. The critical exponents of the weakly disordered Baxter model coincide with those of the weakly disordered Ising model.

Using LEED to study specific heat anomalies of adsorbed overlayers

Near a second order phase boundary the leading thermal dependence of the integrated intensities of “extra” LEED beams is proportional to ∓ | T − T c | 1− α for T ≷ T c where α is the specific heat critical exponent. This behavior occurs when the correlation length exceeds the characteristic length of the instrument but is less than the characteristic size of defect-free regions. We illustrate these ideas using the structure factor of a p(2 × 2) overlayer on a triangular lattice, simulated using Monte Carlo. Similar behavior can appear in other probes of phenomena dependent solely on finite-range correlations.

Using LEED to study specific heat anomalies of adsorbed overlayers

Near a second order phase boundary the leading thermal dependence of the integrated intensities of “extra” LEED beams is proportional to ∓| T− T c| 1-α for T ≷ T c, where α is the specific heat critical exponent. This behavior occurs when the correlation length exceeds the characteristic length of the instrument but is less than the characteristic size of defect-free regions. We illustrate these ideas using the structure factor of a p(2×2) overlayer on a triangular lattice, simulated using Monte Carlo. Similar behavior can appear in other probes of phenomena dependent solely on finite-range correlations.

Critical behaviour of the electrical resistivity in amorphous ferromagnetic alloys

Electrical resistivity ϱ of the amorphous Fe xNi 80−xB 19Si 1 ( x = 10, 13, 16), Fe 20Ni 60B 20 and Fe 20Ni 60P 14B 6 alloys has been measured in a wide temperature range which embraces the second-order magnetic phase transition at the Curie point T C. Analysis of the data reveals that in these systems critical region has a much wider range compared with that in crystalline ferromagnets even though dϱ/dT for T≳T C decreases at a rate that is comparable in both the cases. For the glassy ribbons coming from the same alloy batch, T C and the specific heat critical exponent α have the same values as those determined from our previous magnetic measurements. α for all the investigated alloys possesses values that are typical of a Heisenberg ferromagnet.

Critical Behavior of Birefringence in Two Smectic-A Pentylbenzenethio-Alkoxybenzoates Near the Smectic-C Phase

Abstract The birefringence has been measured in detail in the smectic-A phase of 4-n-pentylbenzenethio-4'-n-octyloxyaniline and 4-n-pentylbenzenethio-4'-n-decyloxyaniline. The reduction of the birefringence near the smectic-C phase has been analyzed in terms of the effect of pretransitional short-range tilt-angle fluctuations. The results show that the two materials exhibit quantitatively similar critical suppression of the birefringence which can be characterized, within the temperature range of the fits, by a specific-heat critical exponent a of 0.15 ± 0.12 and 0.16 ± 0.08, respectively.

Variational principles in renormalization theory

Abstract The variational principles introduced by Kadanoff et al. in the renormalization theory are analyzed. It is shown that the values for the specific heat critical exponent α which can be found by a variational method are restricted to α α = 1 (first order transition). The reason is the confluence of the singularities in the free energy and in the variational parameters. A full implementation of the variational principle changes for the square Ising lattice the earlier obtained α = 0.001756 to α = −0.123413.

Critical behaviour of the magnetic birefringence in (Mn, Zn)F 2

The linear magnetic birefringence (LMB) of Mn 1− c Zn c F 2 was measured between 1.3 K and 300 K. The LMB was found to vary like the magnetic energy where the constant of proportionality is independent of the concentration. Preliminary estimates of the specific heat critical exponents α and α' were obtained for the concentrations in which a phase transition was observed.

Potts model specific heat critical exponents

A series expansion for the free energy of the q-state Potts model on a square lattice is used to estimate the specific heat critical exponents. The analysis is based on a series transformation which was suggested by the known solution of the two-state Potts (Ising) model, and which makes optimum use of the duality theorem. The transformed series is quite smooth. Neville tables yield the estimates alpha (2)=0.0001+or-0.0003 for the two-state model, alpha (3)=0.296+or-0.002 for the three-state model, and alpha (4)=0.45+or-0.02 for the four-state model. The value for alpha (3) differs considerably from one reported by Straley and Fisher (1973), and substantially improves compliance with the Rushbrooke inequality.

Cubic rare-earth compounds: Variants of the three-state Potts model

In appropriate cubic fields rare-earth ions have sixfold degenerate ground states. When the angular momentum of the rare earth is large, the six levels are characterized by states that are directed along the cube edges. Within these states the angular momentum operators ${J}_{x}$, ${J}_{y}$, and ${J}_{z}$ have particularly simple matrix representations. The projection of an isotropic pair coupling between the rare earths onto these sixfold degenerate states leads to an interaction Hamiltonian $\mathcal{H}=\ensuremath{-}\mathcal{I}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{}{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}{\ensuremath{\delta}}_{{l}_{i}{l}_{j}}$, where $\ensuremath{\sigma}$ takes on the values \ifmmode\pm\else\textpm\fi{} 1 and $l$ the values $x$, $y$, and $z$. This interaction is a variant of the three-state Potts model. We add magnetic and quadrupolar anisotropy field terms to the Hamiltonian and determine the symmetry properties of the phase diagram associated with this model. For nonzero quadrupolar anisotropy fields, the model is shown to have the thermodynamic behavior of an Ising model. However, for zero fields we find a new symmetry appears and in the mean-field approximation the model has tricritical-like exponents. This simple model is able to account for the large specific-heat critical exponent ${\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{1}{2}$ which has been observed for holmium antimonide in zero external fields. To the extent that the mean-field approximation is an accurate guide, we predict there are many cubic rare-earth compounds which exhibit tricritical-like behavior in zero field. In addition we find that for pure quadrupole coupling between rare earths in the sixfold degenerate states, the interaction Hamiltonian is exactly the three-state Potts model. In the mean-field approximation this system has a first-order phase transition. However, a small quadrupolar anisotropy field is sufficient to drive the system to a wing critical point. The specific heat has a critical exponent of $\ensuremath{\alpha}=\frac{2}{3} or 1$ depending on the path taken to approach this critical point.